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Math Help - distance function

  1. #1
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    distance function

    Let ρ: RR be a continuous, strictly increasing function (so y>x ρ(y) > ρ(x)). Show that d(x,y) = │ρ(y) ρ(x)│ is a distance function on R, which is complete if and only if limx→∞ρ(x) = ∞ and limx→-∞ρ(x) = -∞.

    here i have to show
    - d is a distance function
    - d is complete if and only if limx→∞ρ(x) = ∞ and limx→-∞ρ(x) = -∞.
    is that right?

    and what does it mean by 'distance function is complete'?
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    What does it mean by 'distance function is complete'?
    A complete metric is a metric in which every Cauchy sequence is convergent.
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  3. #3
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    Oh, i thought the word 'complete' is only for metric space.

    could help me to prove the second part?

    d(x,y) is complete if and only if limx→∞ρ(x) = ∞ and limx→−∞ρ(x) = −∞


    here, how do i define a sequence in the metric ?
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